Can the sorting from an ORDER BY statement be saved explicitly with only one UPDATE statement?

I have this table:

SELECT * FROM items

id | item      | position
---|-----------|----------
 1 | USB cable |        0
 2 | SD card   |        4
 3 | Mouse     |        2
 4 | Keyboard  |        0
 5 | Monitor   |        3

Sorting this table gives this result:

SELECT * FROM items ORDER BY position

id | item      | position
---|-----------|----------
 4 | Keyboard  |        0
 1 | USB cable |        0
 3 | Mouse     |        2
 5 | Monitor   |        3
 2 | SD card   |        4

Now I want to update the table and save the order in the position column:

SELECT * FROM items

id | item      | position
---|-----------|----------
 4 | Keyboard  |        1
 1 | USB cable |        2
 3 | Mouse     |        3
 5 | Monitor   |        4
 2 | SD card   |        5

Can this be done with a single query or do I have to manually loop over all rows and do a manual update?

In case the order is not fully defined (e.g. for USB cable and Keyboard above) I just arbitrarily decided the order.

UPDATE
   T1
SET
   position = T2.rn
FROM
   myTable T1
   JOIN
   (
   SELECT
      id,
      ROW_NUMBER() OVER (ORDER BY position) AS rn
   FROM
      myTable
   ) T2 ON T1.id = T2.id

Note: the order of "Keyboard" and "USB cable" is arbitrary. They both have position = 0

To tie-break positions based on item, add a secondary sort

ROW_NUMBER() OVER (ORDER BY position, item) AS rn

If you have duplicate position,item pairs this will be arbitrary too...

StackExchange Database Administrators Q#5146, answer score: 12